Finite Group Databasebeta

← #13152014

Subgroups — #13152014

Order 1728.

1120 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C3×Dic9C211yesmaximal, characteristic, complemented
2C2×#88662C211yesmaximal, characteristic, complemented
3#11721874C211yesmaximal, characteristic, complemented
4C9×#953C211yesmaximal, characteristic, complemented
5#11721288C212yesmaximal, complemented
6#67×Dic9C212yesmaximal, complemented
7#11721265C211yesmaximal, characteristic, complemented
8#11721255C211yesmaximal, characteristic, complemented
9#11721253C212yesmaximal, complemented
10#11721245C211yesmaximal, characteristic, complemented
11#11721241C212yesmaximal, complemented
12#10595498—31nomaximal
13#10589688—31nomaximal
14C24×C3×C9C2211yescharacteristic
15C2×#6233C2212yes—
16C2×#6233C2211yescharacteristic
17C2×C9×#67C2211yescharacteristic
18C2×C9×#67C2212yes—
19C22×C3×Dic9—24no—
20C22×C3×Dic9C2212yes—
21C22×C3×Dic9C2212yes—
22C22×C3×Dic9C2211yescharacteristic
23C22×C3×Dic9C2211yescharacteristic
24C2×#6195C2211yescharacteristic, complemented
25#88662—24no—
26C2×D3×Dic9C2211yescharacteristic, complemented
27C2×(C3⋊D9)⋊C4C2211yescharacteristic, complemented
28C2×#6189C2211yescharacteristic, complemented
29#88551C2211yescharacteristic
30#88551C2212yes—
31C9×#217C2212yes—
32C9×#217C2211yescharacteristic
33(C2×D3)⋊Dic9C2212yescomplemented
34#88445C2212yescomplemented
35Dic3⋊Dic9C2212yescomplemented
36#88442C2212yescomplemented
37Dic9⋊Dic3C2212yescomplemented
38Dic3×Dic9C2212yescomplemented
39C23×C3×Dic3—31no—
40C2×#3804—31no—
41#64162—31no—
42C3×#953D311yescharacteristic
43#63978—32no—
44Dic3×#67—32no—
45#63966—31no—
46#63964—32no—
47#63961—31no—
48#63955—31no—
49#63952—32no—
50C23×Dic9D311yescharacteristic, complemented
51C2×#3699—31no—
52C9×#120—31no—
53#63515—31no—
54#63515—61no—
55#63501—32no—
56#63497—31no—
57D4×Dic9—32no—
58#63448—31no—
59#611⋊C4—32no—
60#63440—32no—
61#63439—31no—
62C23×C3×C9—22no—
63C23×C3×C9—24no—
64C23×C3×C9C2311yescharacteristic
65C23×C3×C9C2312yes—
66C22×C3⋊D9C2311yescharacteristic
67C22×D3×C9C2311yescharacteristic
68#6233—22no—
69#6233—24no—
70C2×C3⋊Dic9C2311yescharacteristic
71C2×C3⋊Dic9C2312yes—
72C9×#67—22no—
73C9×#67—24no—
74C2×C9×Dic3C2311yescharacteristic
75C2×C9×Dic3C2312yes—
76C2×C3×Dic9—22no—
77C2×C3×Dic9—24no—
78C2×C3×Dic9—24no—
79C2×C3×Dic9D414yescomplemented
80C2×C3×Dic9C2312yes—
81C2×C3×Dic9C2311yescharacteristic
82C2×C3×Dic9C2311yescharacteristic
83#6195—22no—
84D3×Dic9—22no—
85(C3⋊D9)⋊C4—22no—
86#6189—22no—
87#5791—91no—
88#5586—91no—
89C24×C32C2×D311yescharacteristic
90C2×#638—31no—
91C2×#638—32no—
92C2×C3×#67C2×D311yescharacteristic
93C2×C3×#67C2×D312yes—
94C22×C3×Dic3—31no—
95C22×C3×Dic3—32no—
96C22×C3×Dic3—31no—
97C22×C3×Dic3—32no—
98C22×C3×Dic3—64no—
99C2×#627—31no—
100C2×#626—31no—
101C2×#624—31no—
102#3804—64no—
103C2×D3×Dic3—31no—
104C24×C9—21no—
105C24×C9C2×D311yescharacteristic
106#3757—31no—
107#3757—32no—
108C3×#217C2×D312yes—
109C3×#217C2×D311yescharacteristic
110#3724—32no—
111Dic3⋊Dic3—32no—
112Dic3⋊Dic3—32no—
113#3722—32no—
114(C2×D3)⋊Dic3—32no—
115Dic32—32no—
116C2×D4×C9—32no—
117C2×D4×C9—31no—
118C2×#611—32no—
119C2×#611—31no—
120C2×#611—31no—
121C2×#611—62no—
122C2×#611—61no—
123C22×Dic9—24no—
124C22×Dic9—31no—
125C22×Dic9C2×D312yes—
126C22×Dic9C2×D311yescharacteristic
127C22×Dic9C2×D312yes—
128C22×Dic9C2×D311yescharacteristic
129#3699—64no—
130C2×C4×D9—31no—
131C2×Dic18—31no—
132C9×C22⋊C4—31no—
133C9×C22⋊C4—32no—
134#3676—32no—
135#3676—31no—
136#3676—32no—
137#3676—61no—
138#3676—62no—
139(C2×D9)⋊C4—32no—
140C4⋊Dic9—32no—
141Dic9⋊C4—32no—
142Dic9⋊C4—32no—
143C4×Dic9—32no—
144C22×C3×C9—28no—
145C22×C3×C9—22no—
146C22×C3×C9—22no—
147C22×C3×C9—22no—
148C22×C3×C9C2×D412yes—
149C22×C3×C9#4114yescomplemented
150C22×C3×C9C2411yescharacteristic
151C2×C3⋊D9—21no—
152C2×C3⋊D9—22no—
153C2×D3×C9—21no—
154C2×D3×C9—22no—
155C3⋊Dic9—21no—
156C3⋊Dic9—22no—
157C9×Dic3—21no—
158C9×Dic3—22no—
159C3×Dic9—22no—
160C3×Dic9C2×D412yes—
161C3×Dic9C2×D412yes—
162C23×C3×C4—91no—
163C23×Dic3—31no—
164C2×#237—91no—
165C2×#235—91no—
166C3×#120—31no—
167#953—91no—
168#953—91no—
169#953—181no—
170#953D911yescharacteristic, complemented
171#939—92no—
172#935—91no—
173D4×Dic3—92no—
174#930—92no—
175#929—91no—
176#929—91no—
177C4×#67—92no—
178C4×#67—92no—
179#926—91no—
180#924—92no—
181#886—91no—
182#67⋊C4—92no—
183#878—92no—
184#877—91no—
185C23×C32—22no—
186C23×C32—24no—
187C23×C32C22×D311yescharacteristic
188C23×C32C22×D312yes—
189C22×C3⋊D3—31no—
190C22×C3×D3C22×D311yescharacteristic
191#638—62no—
192#638—64no—
193C2×C3⋊Dic3—32no—
194C2×C3⋊Dic3—31no—
195C3×#67—22no—
196C3×#67—24no—
197C2×C3×Dic3—34no—
198C2×C3×Dic3—31no—
199C2×C3×Dic3—31no—
200C2×C3×Dic3—32no—
201C2×C3×Dic3—64no—
202C2×C3×Dic3—64no—
203C2×C3×Dic3—62no—
204C2×C3×Dic3C22×D312yes—
205C2×C3×Dic3C22×D311yescharacteristic
206#627—62no—
207#626—62no—
208#624—62no—
209D3×Dic3—62no—
210C23×C9—24no—
211C23×C9—22no—
212C23×C9—22no—
213C23×C9—21no—
214C23×C9—31no—
215C23×C9—44no—
216C23×C9—42no—
217C23×C9C22×D312yes—
218C23×C9C22×D311yescharacteristic
219C22×D9—31no—
220C22×D9—61no—
221D4×C9—64no—
222D4×C9—62no—
223C2×C4×C9—32no—
224C2×C4×C9—31no—
225#611—64no—
226#611—62no—
227#611—62no—
228#611—124no—
229#611—122no—
230C2×Dic9—22no—
231C2×Dic9—24no—
232C2×Dic9—24no—
233C2×Dic9—32no—
234C2×Dic9—31no—
235C2×Dic9—62no—
236C2×Dic9—61no—
237C2×Dic9—62no—
238C2×Dic9#6714yescomplemented
239C2×Dic9C22×D312yes—
240C2×Dic9C22×D311yescharacteristic
241C2×Dic9C22×D311yescharacteristic
242C4×D9—62no—
243Dic18—62no—
244#525—271noSylow
245C2×C3×C9—22no—
246C2×C3×C9—24no—
247C2×C3×C9C22×D411yescharacteristic
248C2×C3×C9C2×#4112yes—
249C3⋊D9—41no—
250D3×C9—41no—
251C24×C3—21no—
252C24×C3C2×D911yescharacteristic
253C24×C3D3211yescharacteristic
254C2×C3×D4—32no—
255C2×C3×D4—31no—
256C22×C3×C4—91no—
257C22×C3×C4—92no—
258C22×C3×C4—92no—
259C22×C3×C4—91no—
260C22×C3×C4—184no—
261C2×#67—91no—
262C2×#67—92no—
263C2×#67—91no—
264C2×#67—92no—
265C2×#67—91no—
266C2×#67—182no—
267C2×#67—181no—
268C2×#67C2×D912yes—
269C2×#67C2×D911yescharacteristic
270C22×Dic3—31no—
271C22×Dic3—32no—
272C22×Dic3—32no—
273C22×Dic3—31no—
274C22×Dic3—64no—
275C22×Dic3—91no—
276#237—184no—
277#235—184no—
278C2×D12—91no—
279C2×C4×D3—91no—
280C2×C4×D3—91no—
281C2×C4×D3—91no—
282C2×Dic6—91no—
283C2×Dic6—91no—
284C3×C22⋊C4—31no—
285C3×C22⋊C4—32no—
286#217—92no—
287#217—92no—
288#217—91no—
289#217—91no—
290#217—92no—
291#217—181no—
292#217—182no—
293#217C2×D912yes—
294#217C2×D911yescharacteristic
295(C2×D3)⋊C4—92no—
296(C2×D3)⋊C4—92no—
297(C2×D3)⋊C4—92no—
298C4⋊Dic3—92no—
299C4⋊Dic3—92no—
300Dic3⋊C4—92no—
301Dic3⋊C4—92no—
302Dic3⋊C4—92no—
303Dic3⋊C4—92no—
304C4×Dic3—92no—
305C4×Dic3—92no—
306C22×C32—28no—
307C22×C32—22no—
308C22×C32—22no—
309C22×C32—22no—
310C22×C32D3×D412yes—
311C22×C32#23714yes—
312C22×C32C23×D311yescharacteristic
313C2×C3⋊D3—61no—
314C2×C3⋊D3—62no—
315C2×C3×D3—21no—
316C2×C3×D3—22no—
317C3⋊Dic3—61no—
318C3⋊Dic3—62no—
319C3×Dic3—21no—
320C3×Dic3—22no—
321C3×Dic3—32no—
322C3×Dic3—32no—
323C3×Dic3—62no—
324C22×C9—28no—
325C22×C9—22no—
326C22×C9—22no—
327C22×C9—22no—
328C22×C9—21no—
329C22×C9—24no—
330C22×C9—22no—
331C22×C9—42no—
332C22×C9—42no—
333C22×C9—42no—
334C22×C9—48no—
335C22×C9—62no—
336C22×C9—61no—
337C22×C9#23514yescomplemented
338C22×C9C2×#6712yes—
339C22×C9C23×D311yescharacteristic
340C2×D9—62no—
341C2×D9—61no—
342C2×D9—122no—
343C2×D9—121no—
344C4×C9—62no—
345C4×C9—61no—
346Dic9—22no—
347Dic9—62no—
348Dic9—61no—
349Dic9—122no—
350Dic9—121no—
351Dic9C2×#6712yes—
352Dic9C2×#6712yes—
353C2×#41—271no—
354C23×C4—91no—
355#123—271no—
356#123—271no—
357C22⋊Q8—272no—
358#121—272no—
359#120—31no—
360#120—271no—
361C4×D4—272no—
362C4×D4—272no—
363#117—271no—
364C3×C9#52511yescharacteristic, complemented, Sylow
365C23×C3—22no—
366C23×C3—24no—
367C23×C3—24no—
368C23×C3—22no—
369C23×C3—22no—
370C23×C3—21no—
371C23×C3—31no—
372C23×C3—44no—
373C23×C3—42no—
374C23×C3C22×D911yescharacteristic
375C23×C3C22×D912yes—
376C23×C3C2×D3212yes—
377C23×C3C2×D3211yescharacteristic
378C22×D3—91no—
379C22×D3—91no—
380C22×D3—181no—
381C22×D3C22×D911yescharacteristic
382C3×D4—64no—
383C3×D4—62no—
384C2×C3×C4—32no—
385C2×C3×C4—31no—
386C2×C3×C4—94no—
387C2×C3×C4—91no—
388C2×C3×C4—91no—
389C2×C3×C4—92no—
390C2×C3×C4—184no—
391C2×C3×C4—184no—
392C2×C3×C4—182no—
393#67—22no—
394#67—24no—
395#67—182no—
396#67—182no—
397#67—184no—
398#67—184no—
399#67—182no—
400#67—364no—
401#67—362no—
402C2×Dic3—31no—
403C2×Dic3—31no—
404C2×Dic3—34no—
405C2×Dic3—32no—
406C2×Dic3—64no—
407C2×Dic3—64no—
408C2×Dic3—62no—
409C2×Dic3—92no—
410C2×Dic3—91no—
411C2×Dic3—92no—
412C2×Dic3—91no—
413C2×Dic3—182no—
414C2×Dic3—181no—
415C2×Dic3—182no—
416C2×Dic3C22×D912yes—
417C2×Dic3C22×D911yescharacteristic
418D12—182no—
419C4×D3—182no—
420C4×D3—182no—
421C4×D3—182no—
422Dic6—182no—
423Dic6—182no—
424C2×C32—22no—
425C2×C32—24no—
426C2×C32C2×D3×D411yescharacteristic
427C2×C32C2×#23712yes—
428C3⋊D3—121no—
429C3×D3—41no—
430C2×C9—24no—
431C2×C9—22no—
432C2×C9—22no—
433C2×C9—21no—
434C2×C9—44no—
435C2×C9—42no—
436C2×C9—121no—
437C2×C9C2×#23512yes—
438C2×C9C22×#6711yescharacteristic
439D9—121no—
440D9—241no—
441C24D3×D911yescharacteristic
442#41—544no—
443C2×Q8—271no—
444C2×D4—32no—
445C2×D4—31no—
446C2×D4—271no—
447C2×D4—272no—
448C2×D4—271no—
449C22×C4—91no—
450C22×C4—92no—
451C22×C4—92no—
452C22×C4—91no—
453C22×C4—184no—
454C22×C4—271no—
455C22×C4—271no—
456C4⋊C4—272no—
457C4⋊C4—272no—
458C4⋊C4—272no—
459C22⋊C4—31no—
460C22⋊C4—32no—
461C22⋊C4—272no—
462C22⋊C4—272no—
463C22⋊C4—271no—
464C22⋊C4—272no—
465C42—272no—
466C22×C3—28no—
467C22×C3—28no—
468C22×C3—22no—
469C22×C3—22no—
470C22×C3—22no—
471C22×C3—22no—
472C22×C3—22no—
473C22×C3—22no—
474C22×C3—21no—
475C22×C3—24no—
476C22×C3—22no—
477C22×C3—42no—
478C22×C3—42no—
479C22×C3—42no—
480C22×C3—48no—
481C22×C3—62no—
482C22×C3—61no—
483C22×C3D4×D912yes—
484C22×C3#369914yescomplemented
485C22×C3C23×D911yescharacteristic
486C22×C3#380414yes—
487C22×C3D3×#6712yes—
488C22×C3C22×D3211yescharacteristic
489C2×D3—21no—
490C2×D3—22no—
491C2×D3—182no—
492C2×D3—181no—
493C2×D3—181no—
494C2×D3—182no—
495C2×D3—362no—
496C2×D3—361no—
497C3×C4—62no—
498C3×C4—61no—
499C3×C4—92no—
500C3×C4—92no—
501C3×C4—182no—
502Dic3—21no—
503Dic3—22no—
504Dic3—32no—
505Dic3—32no—
506Dic3—62no—
507Dic3—182no—
508Dic3—181no—
509Dic3—181no—
510Dic3—182no—
511Dic3—362no—
512Dic3—361no—
513C32#558611yescharacteristic
514C9—21no—
515C9#579111yescharacteristic, complemented
516C23—24no—
517C23—22no—
518C23—31no—
519C23—271no—
520C23C2×D3×D912yes—
521C23C2×D3×D911yescharacteristic
522Q8—542no—
523D4—64no—
524D4—62no—
525D4—542no—
526D4—544no—
527D4—542no—
528C2×C4—32no—
529C2×C4—31no—
530C2×C4—91no—
531C2×C4—91no—
532C2×C4—94no—
533C2×C4—92no—
534C2×C4—184no—
535C2×C4—184no—
536C2×C4—182no—
537C2×C4—272no—
538C2×C4—271no—
539C2×C4—542no—
540C2×C4—542no—
541C2×C3—24no—
542C2×C3—22no—
543C2×C3—22no—
544C2×C3—24no—
545C2×C3—22no—
546C2×C3—21no—
547C2×C3—44no—
548C2×C3—42no—
549C2×C3—121no—
550C2×C3C2×D4×D911yescharacteristic
551C2×C3C2×#369912yes—
552C2×C3C2×#380412yes—
553C2×C3C2×D3×#6711yescharacteristic
554D3—41no—
555D3—361no—
556D3—361no—
557D3—721no—
558C22—28no—
559C22—22no—
560C22—22no—
561C22—22no—
562C22—62no—
563C22—61no—
564C22—542no—
565C22—541no—
566C22#8866214yescomplemented
567C22D9×#6712yes—
568C22C22×D3×D911yescharacteristic
569C4—62no—
570C4—61no—
571C4—92no—
572C4—92no—
573C4—182no—
574C4—542no—
575C4—541no—
576C3—21no—
577C3#1058968811yescharacteristic, complemented
578C3#1059549811yescharacteristic
579C2—24no—
580C2—22no—
581C2—121no—
582C2—1081no—
583C2C2×#8866212yes—
584C2C2×D9×#6711yescharacteristic
subgroupG/H
centerC22C22×D3×D9
commutator, cosocleC22×C3×C9C24
FrattiniC22×C3C22×D32
FittingC24×C3×C9C22
socleC22×C32C23×D3
hypercenterC24D3×D9
hypocenterC3×C9#525
LCSG ▷ C22×C3×C9 ▷ C3×C9
UCSC1 ◁ C22 ◁ C24

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C23×C3×Dic9C2semidirect
2. C2×#88662C2semidirect
3. #11721874C2semidirect
4. C9×#953C2semidirect
5. #11721288C2semidirect
6. #67×Dic9C2semidirect
7. #11721265C2semidirect
8. #11721255C2semidirect
9. #11721253C2semidirect
10. #11721245C2semidirect
11. #11721241C2semidirect
24. C2×#6195C22semidirect
26. C2×D3×Dic9C22semidirect
27. C2×(C3⋊D9)⋊C4C22semidirect
28. C2×#6189C22semidirect
33. (C2×D3)⋊Dic9C22semidirect
34. #88445C22semidirect
35. Dic3⋊Dic9C22semidirect
36. #88442C22semidirect
37. Dic9⋊Dic3C22semidirect
38. Dic3×Dic9C22semidirect
50. C23×Dic9D3semidirect
79. C2×C3×Dic9D4semidirect
149. C22×C3×C9#41semidirect
170. #953D9semidirect
238. C2×Dic9#67semidirect
337. C22×C9#235semidirect
364. C3×C9#525semidirect
484. C22×C3#3699semidirect
515. C9#5791semidirect
566. C22#88662semidirect
577. C3#10589688semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C23×C3×Dic92
C2×#886622
#117218742
C9×#9532
#117212882
#67×Dic92
#117212652
#117212552
#117212532
#117212452
#117212412
#105954983
#105896883